Finite Group Databasebeta

← #22600568

Subgroups — #22600568

Order 9088.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986723C211yesmaximal, characteristic, complemented
2D4×D284C211yesmaximal, characteristic
3#17986347C211yesmaximal, characteristic, complemented
4#17986326C212yesmaximal, complemented
5#17986246C212yesmaximal, complemented
6#17986200C211yesmaximal, characteristic, complemented
7#17986151C212yesmaximal, complemented
8C4×#11765853C211yesmaximal, characteristic, complemented
9#17986076C211yesmaximal, characteristic, complemented
10#17986002C212yesmaximal, complemented
11#474×C71C211yesmaximal, characteristic, complemented
12C22×D284—21no—
13C2×D4×D71C2212yes—
14C2×#11765854C2212yescomplemented
15C2×#11765853C2211yescharacteristic
16C2×#11765848C2212yescomplemented
17#14556125C2211yescharacteristic
18#14556123—22no—
19#14556117—21no—
20#14556108C2212yescomplemented
21#14556103C2212yes—
22#14556102—21no—
23#14556098—22no—
24Q8×Dic71—21no—
25C4×Dic142C2211yescharacteristic, complemented
26C4×D284C2211yescharacteristic
27#14556086C2212yescomplemented
28#14556059C2211yescharacteristic, complemented
29#14556051C2211yescharacteristic, complemented
30#14556050C2212yescomplemented
31#14556049C2211yescharacteristic
32#14556048C2212yescomplemented
33#14556048C2212yescomplemented
34(C2×D71)⋊C8C2212yescomplemented
35C4×C71⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C71C2211yescharacteristic, complemented
37C4×D4×C71C2211yescharacteristic
38#107×C71C2212yescomplemented
39C4⋊C8×C71C2211yescharacteristic, complemented
40#102×C71C2212yes—
41C23×D71—22no—
42C2×Dic142C2311yescharacteristic
43C2×D284—22no—
44C2×D284—21no—
45C2×D284—21no—
46C2×D284—44no—
47C2×D284D411yescharacteristic
48C2×D284C2312yes—
49C2×C22⋊D71—22no—
50D4×D71—24no—
51D4×D71—42no—
52C2×C4×D71C2312yes—
53#11765854—42no—
54#11765853—22no—
55#11765848—42no—
56C2×C71⋊C8C2311yescharacteristic
57C2×C71⋊C8C2311yescharacteristic
58C4⋊Dic71—22no—
59C4⋊Dic71D411yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4D412yes—
66C4×Dic71—21no—
67C4×Dic71—22no—
68C2×Q8×C71—21no—
69C2×D4×C71C2311yescharacteristic
70C22×C4×C71—21no—
71C2×C8×C71C2312yes—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2312yes—
74C4⋊C4×C71C2311yescharacteristic
75C22⋊C4×C71—21no—
76C42×C71C2311yescharacteristic
77C23×C71—21no—
78C22×D71—22no—
79C22×D71—44no—
80C22×D71—44no—
81C22×D71—82no—
82C22×D71C2×D412yes—
83Q8×C71—42no—
84D4×C71—21no—
85D4×C71#4112yes—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—42no—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2411yescharacteristic
92C22⋊D71—44no—
93C2×Dic71—21no—
94C2×Dic71—21no—
95C2×Dic71C2×D412yes—
96D284—22no—
97D284—22no—
98D284—22no—
99D284—44no—
100D284—42no—
101D284—81no—
102C4×D71—22no—
103C4×D71—22no—
104Dic142—21no—
105Dic142—21no—
106C8×C71—22no—
107C71⋊C8—21no—
108C71⋊C8—21no—
109C22×C71—22no—
110C22×C71—41no—
111C22×C71—41no—
112C22×C71C22×D411yescharacteristic
113C2×D71—22no—
114C2×D71—22no—
115C2×D71—22no—
116C2×D71—42no—
117C2×D71—42no—
118C2×D71—42no—
119C2×D71—88no—
120C4×C71—21no—
121C4×C71—41no—
122C4×C71—42no—
123C4×C71AGL(1,ℤ/8)12yes—
124C4×C71C2×#4111yescharacteristic
125C4×C71H(5,𝔽2)11yescharacteristic
126Dic71—22no—
127Dic71—41no—
128Dic71—41no—
129C2×C71—42no—
130C2×C71#54611yescharacteristic
131C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C71#57711yescharacteristic
133D71—42no—
134D71—82no—
135#3324—711nomaximal, Sylow
136C71#332411yescharacteristic, complemented, Sylow
137#556—711no—
138D42—711no—
139#486—711no—
140#482—712no—
141#474D7111yescharacteristic, complemented
142#468—712no—
143#460—711no—
144#450—712no—
145C4×SD16—711no—
146#436—711no—
147#418—712no—
148C22×D4—712no—
149C22×D4—1421no—
150C2×SD16—711no—
151C2×SD16—712no—
152C2×M16—712no—
153C4⋊Q8C2×D7111yescharacteristic, complemented
154#127—711no—
155#126—1422no—
156#124—1421no—
157C22⋊Q8—712no—
158#121—712no—
159#121—1421no—
160#120—1422no—
161C4×Q8—711no—
162C4×Q8—1421no—
163C4×D4—711no—
164C4×D4C2×D7111yescharacteristic
165#117—712no—
166#107C2×D7112yescomplemented
167#106—711no—
168C4⋊C8C2×D7111yescharacteristic, complemented
169#103—711no—
170#102—711no—
171#102—712no—
172#102—712no—
173#102—712no—
174#102C2×D7112yes—
175C22⋊C8—712no—
176C4×C8—711no—
177C24—1422no—
178C2×Q8—21no—
179C2×Q8—711no—
180C2×D4—2844no—
181C2×D4—2842no—
182C2×D4—711no—
183C2×D4—712no—
184C2×D4—1422no—
185C2×D4—1422no—
186C2×D4—1424no—
187C2×D4—1421no—
188C2×D4—1421no—
189C2×D4C22×D7111yescharacteristic
190C22×C4—21no—
191C22×C4—712no—
192SD16—2842no—
193SD16—1422no—
194M16—2842no—
195C2×C8—711no—
196C2×C8—711no—
197C2×C8C22×D7112yes—
198C4⋊C4—21no—
199C4⋊C4—711no—
200C4⋊C4—711no—
201C4⋊C4—1422no—
202C4⋊C4—1421no—
203C4⋊C4C22×D7112yes—
204C4⋊C4C22×D7111yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—712no—
207C22⋊C4—1422no—
208C22⋊C4—1422no—
209C22⋊C4—1422no—
210C42—1422no—
211C42—1421no—
212C42C22×D7111yescharacteristic
213C23—21no—
214C23—2844no—
215C23—2844no—
216C23—5682no—
217C23—712no—
218C23—1422no—
219Q8—42no—
220Q8—1421no—
221Q8—1421no—
222D4—21no—
223D4—2844no—
224D4—2844no—
225D4—2842no—
226D4—5681no—
227D4—1422no—
228D4—1422no—
229D4—1422no—
230D4#1176586612yes—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—712no—
235C2×C4—1421no—
236C2×C4—1421no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4D4×D7111yescharacteristic
240C2×C4D4×D7111yescharacteristic
241C2×C4C23×D7111yescharacteristic
242C8—22no—
243C8—1421no—
244C8—1421no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—2842no—
249C22—2842no—
250C22—2842no—
251C22—5688no—
252C22—1422no—
253C22—1422no—
254C22—1422no—
255C22C2×D4×D7111yescharacteristic
256C4—21no—
257C4—41no—
258C4—42no—
259C4—2841no—
260C4—2841no—
261C4—1422no—
262C4#1455615212yes—
263C4C2×#1176586611yescharacteristic
264C4#1455617411yescharacteristic
265C2—42no—
266C2—2842no—
267C2—5682no—
268C2#1798664011yescharacteristic
269C2C2×#1455615211yescharacteristic
270C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#474×C71C2
socleC22×C71C22×D4
hypercenter#474D71
hypocenterC71#3324
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17986723C2semidirect
3. #17986347C2semidirect
4. #17986326C2semidirect
5. #17986246C2semidirect
6. #17986200C2semidirect
7. #17986151C2semidirect
8. C4×#11765853C2semidirect
9. #17986076C2semidirect
10. #17986002C2semidirect
11. #474×C71C2semidirect
14. C2×#11765854C22semidirect
16. C2×#11765848C22semidirect
20. #14556108C22semidirect
25. C4×Dic142C22semidirect
27. #14556086C22semidirect
28. #14556059C22semidirect
29. #14556051C22semidirect
30. #14556050C22semidirect
32. #14556048C22semidirect
33. #14556048C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. C4×C71⋊C8C22semidirect
36. C4⋊Q8×C71C22semidirect
38. #107×C71C22semidirect
39. C4⋊C8×C71C22semidirect
136. C71#3324semidirect
141. #474D71semidirect
153. C4⋊Q8C2×D71semidirect
166. #107C2×D71semidirect
168. C4⋊C8C2×D71semidirect